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equation of a circle

 Definition/Summary A circle has many definitions, a classical one being "the locus of all points on a plane that are equidistant from a given point, which is referred to as the 'center' of the circle".

 Equations Equation for a circle with it's center as the origin and radius 'r': $$x^2 + y^2 = r^2$$ Equation for a circle with center at (a, b) and radius 'r': $$(x - a)^2 + (y - b)^2 = r^2$$ General equation (expanded) for a circle: $$x^2 + y^2 + 2fx + 2gy + c = 0$$ The center of such a circle is given as (-f, -g) and the radius 'r' is given by: $$r = \sqrt{g^2 + f^2 - c}$$ The slope of the tangent at a point 'x' for a circle centered at (a, b) is given as: $$\tan \theta = -\frac{x - a}{y - b}$$ Area of a circle is given as: $$A = \pi r^2$$ Length of an arc subtending an angle $\theta$ is given as: $$L = r\theta$$ For the length of the complete circle, $\theta = 2\pi$ For a circle, centered at origin, and radius 'r' a point on the circle, the radius to which makes an angle $\theta$ with the positive x-axis is given as: $$x = r\sin \theta$$ $$y = r\cos \theta$$

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 Recent forum threads on equation of a circle

 Breakdown Mathematics > Geometry >> Coordinate Geometry

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 Extended explanation Other definitions from more general concepts: 1. a special instance of an ellipse, having an eccentricity, $e = 0$ (i.e., equal lengths of major and minor axes), 2. a conic section formed by the intersection of a cone with a plane normal to its axis of symmetry, 3. the two-dimensional instance of an n-dimensional hypersphere.

Commentary

 tiny-tim @ 03:55 PM Jan12-09 changed title from "circle (coordinate geometry)", since that was getting no autolinking, to "equation of a circle", since forum searches showed that that got 50% more autolinks than "equation of the circle", and three times as many as "equation of the circle"

 tiny-tim @ 09:31 AM May16-08 This being in the Coordinate Geometry section, I think the title should be "Circle (Coordinate Geometry)", making it clear to both readers and contributors what it limits are. That leaves the way clear for someone else to create a "Circle (Plane Geometry)" entry in the Plane Geometry section.

 robphy @ 05:46 AM May14-08 A circle is a one-dimensional sphere (hence the symbol [ctex]S^1[/ctex]).

 Hootenanny @ 01:23 PM May13-08 @rohanprabhu: Done.

 rohanprabhu @ 12:41 PM May13-08 @RandallBSmith: thanks for pointing out the mistake. Second mistake is in the general equation of the circle... It should be 2gx + 2fy instead of 2gx + 2fx.. i missed the 'y' term and mistook it with an 'x'. I would be grateful if you could make those corrections.

 Gokul43201 @ 10:41 AM May13-08 Fixed, thanks.

 RandallBSmith @ 09:39 AM May13-08 Just visiting this site for the first time,there seems to be an error in the length of an arc subtending an angle theta. (Theta should be multiplied by r, not divided by)